Mathematics for the interested outsider

Row- and Column-Stabilizers

Every Young tableau with shape gives us two subgroups of , the “row-stabilizer” and the “column-stabilizer” . These are simple enough to define, but to write them succinctly takes a little added flexibility to our notation.

Given a set , we’ll write for the group of permutations of that set. For instance, the permutations that only mix up the elements of the set make up

Now, let’s say we have a tableau with rows . Any permutation that just mixes up elements of leaves all but the first row alone when acting on . Since it leaves every element on the row where it started, we say that it stabilizes the rows of . These permutations form the subgroup . Of course, there’s nothing special about here; the subgroups also stabilize the rows of . And since entries from two different subgroups commute, we’re dealing with the direct product:

We say that is the row-stabilizer subgroup, since it consists of all the permutations that leave every entry in on the row where it started. Clearly, this is the stabilizer subgroup of the Young tabloid.

The column-stabilizer is defined similarly. If has columns , then we define the column-stabilizer subgroup

Now column-stabilizers do act nontrivially on the tabloid . The interaction between rearranging rows and columns of tableaux will give us the representations of we’re looking for.

[…] row of . They cannot be in the same column of , since if they were then the swap would be in the column-stabilizer . Then we could conclude that , which we assumed not to be the case. But if no two entries from the […]

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This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

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